There is a function $f: \mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$.
$f$ satisfies the following property for any $x\neq x'$ and $y\neq y'$: $$f(x,y')>f(x,y)~\textrm{implies}~f(x',y')\geq f(x,y')~\textrm{and}~f(x',y')\geq f(x',y) .\tag{1}$$
Is there a name for this property? or Is there any stronger and well-known property that guarantees this property?
We may look at the condition as a combination of two conditions: $$f(x,y')>f(x,y)~\textrm{implies}~f(x',y')\geq f(x,y')~\textrm{and}\tag{2}$$ $$f(x,y')>f(x,y)~\textrm{implies}~f(x',y')\geq f(x',y) .\tag{3}$$ If there is no known name for the combined property, namely (1), would there be a name for (2) or (3), or what kind of functional property on $f$ would imply (1), (2) or (3)?
This is not an answer to what you asked for. I think your function must be somewhat special, and would like to see it, if you know it, before saying anything.
Suppose $f(x, y')\color{\red}{\ge} f(x, y)$ implies $f(x', y')\ge\max(f(x, y'), f(x', y)).$ In particular, it says $f(x', y')\ge f(x', y).$ So, we must have $f(x, y')\ge\max(f(x', y'), f(x, y)),$ and consequently $$f(x', y')=f(x, y')\qquad\forall x, x'.$$ So, in this case, $f$ depends only on the second coordinate.